Introduction

Once you've mastered adding (and subtracting), you're then taught multiplication, which is really just a fast way to add the same number over and over again. Adding can teach you to answer 9 + 9 + 9 + 9, but multiplication turns that problem into 9 * 4, which is easier to write and, once you've mastered multiplication, quicker to solve.

What multiplication is to adding, exponents are to multiplication. With multiplication, you could write out 9 * 9 * 9 * 9, but exponents give you a shorthand notation for that process, writing it as 94. When you write out an exponential equation in a manner like xy, x is referred to as the base, and y is referred to as the exponent.

Is it really possible to work with exponents in your head? Solomon W. Golomb (This and all other external links in this tutorial will open in a new window) often astounds people by performing seemingly complex exponential problems in his head. In one famous incident, his college biology teacher explained that humans had 24 chromosomes (as was believed at the time), so the number of possibilities was 224, and jokingly added that everyone knew what number that was. Young Mr. Golomb (now Dr. Golomb) immediately replied that it was 16,777,216. The teacher thought he was trying to be funny, performed the calculation on a calculator, only to find out that 224 was indeed 16,777,216!

How?

The answer is part pattern recognition, part memorization, and part math. To start, we need to learn the answers to all the exponential expressions XY, where both X and Y range from 0 to 10. You might think of these as your exponential tables, as opposed to your multiplication tables.

The thought of memorizing the answers to 121 exponential expressions (11 numbers from 0 to 10 taken to any of the 11 powers from 0 to 10) may sound tough, but before we even begin memorizing answers, we'll start with a few easy-to-remember patterns to cut down on that number.

Base of 0

Taking 0 to any power from 1 to 10 is simple, the answer is 0! 00, however, is a special case. Depending on your source, it's either 0, 1, or undefined. If you're having someone verify this with a calculator, just respond with ERROR.

Base of 10

Taking 10 to any power from 0 to 10 is easy, as we all learned in school. Write down a 1, look at the exponent, and write down that many zeroes after it! 102? That's a 1 followed by 2 zeroes, or 100. 109? 1,000,000,000 (1 followed by 9 zeroes). You'll probably want to remember that 103 is a thousand, 106 is a million, and 109 is one billion.

Exponential Equation

Answer to Equation

100

1

101

10

102

100

103

1,000

104

10,000

105

100,000

106

1,000,000

107

10,000,000

108

100,000,000

109

1,000,000,000

1010

10,000,000,000

Exponent of 0

For any number from 1 to 10 to the power of 0, the answer is always 1 (Why?). As discussed above, 00 is an unusual case.

Exponential Equation

Answer to Equation

00

Indeterminate

10

1

20

1

30

1

40

1

50

1

60

1

70

1

80

1

90

1

100

1

Exponent of 1

First powers are also very easy. Any number to the first power is itself. 51? 5! 91? 9! 4,326,5281? 4,326,528!

Exponential Equation

Answer to Equation

01

0

11

1

21

2

31

3

41

4

51

5

61

6

71

7

81

8

91

9

101

10

The Remaining Exponents

With just those few simple patterns above, you've already learned 49 different answers! There's still 72 to go, but that number will quickly be minimized.

Squares and Cubes

Next, you'll need to learn the numbers to the 2nd and 3rd powers. These should be learned so that you know them cold. Not only will knowing these by heart help when giving the answers, but they will also help you handle larger equations later on.

Exponent of 2

Taking a number to the 2nd power is also known as squaring it. If you're still comfortable with your times tables up to 10 times 10, and can recall that 82 is 64 and that 92 is 81, you don't need to memorize the squares. For those who do need to memorize the squares, I've given them to you in the chart below. Since you've learned the answers for 02, 12, and 102 on the previous page, I won't include those in the chart below.

Exponential Equation

Answer to Equation

22

4

32

9

42

16

52

25

62

36

72

49

82

64

92

81

Exponent of 3

Taking a number to the 3rd power is also referred to as cubing a number. It's less common to know the cubes offhand than it is for the squares.

Those of you who've put the time in to learn the root extraction feat will have an advantage, as they will already know the cubes by heart! For those who haven't memorized them already, here they are:

Exponential Equation

Answer to Equation

23

8

33

27

43

64

53

125

63

216

73

343

83

512

93

729

Learning the Squares and Cubes

To help make sure you remember the squares and cubes, go to the exponent quiz page, and click on the Squares and Cubes button to practice them. Once you have these memorized, you've only got 56 more to go. This means you've already memorized more than half of the 121 answers!

Learning the Remaining Exponents

Before we continue on, you will need to practice with the Link System and the Major System, in order to commit these answers to memory.

4th and 5th Powers

Prerequisites:

To convert these exponential exponential expressions for use with the Major System, each equation will be converted into a 2-digit number, with the base in the 10s place, and the exponent as the 1s place. For example, to remember the problem for 64, think of it as 64, and use the mnemonic you developed for 64 (I use JaR). The answer to 64 is 1,296, which translates into hiD hoNey PouCH. Use the link system to link JaR to the phrase hiD hoNey PouCH, and the Major System to convert those words and phrase into numbers. Properly done, given a problem like 64, your thought process should go something like this: 64 = 64 = JaR = hiD hoNey PouCH = 1,296

Does this mean that, for a problem like 610, you'll need to develop a mnemonic for 610? No. To make things easier, all bases to the 10th power will represented by a 0 in the 1s place. For 610, you'll use your mnemmonic for 60, for 710, you'll use your mnemmonic for 70, and so on. Other than this exception, 10th powers are handled like all the other numbers. If you've practiced my 400 digits of Pi feat, this will be familiar to you.

In all the charts, I'll be including the exponential equation, the key number to use for the mnemonic, the equation answer, the key number mnemonic, and the answer mnemonic. If you've developed mnemonic for the numbers from 1-100 different than the ones I employ in this tutorial, feel free to use those. The important thing is to make sure that you're able to link them to the answer mnemonic, so that you get the correct answer.

Occasionally, you'll see words in the mnemonics highlighted. These are ones that I believe not everybody will be familiar with, and lead to Wikipedia links explaining exactly what they mean. All these links will open up in a new window.

As the numbers get larger and the charts have more columns, you may find that it's more difficult to read the mnemonics. If so, you can simply re-adjust the widths of the columns in the tables below by clicking on the dividing line and dragging it left or right as desired.

Exponent of 4

Unlike taking a number to the 2nd or 3rd power, taking a number to a 4th or higher power don't have any commonly accepted name like squaring or cubing. They're simply referred to as taking the number to the given power (e.g., 4th power, 5th power).

Learning Up to the 8th Power

It's time for another break. To make sure you have all the numbers up to the 8th power memorized, and to reinforce all the ones you've learned previously, go to the exponent quiz page, and click on the 6th, 7th, and 8th Powers button to practice them.